Thermal Hall Effect. We studied the topological magnon Hall effect using the non-equilibrium Green's function (NEGF) formalism, see Fig. REF . Calculation details are provided in the Supplementary Information. A temperature gradient was introduced to a finite sample to study the magnon heat current along the \(x\)  -direction. The generation of Berry curvature (due to the presence of DM interaction) leads to a current deflection in the \(y\)  -direction. This results in chiral edge current transport. The chiral current is localized. For a selected energy level in the gap, the orientation of the chiral current corresponds well with the sign of the winding number for the edge states as shown in Fig. REF a. With a temperature gradient along the \(x\) -direction, there is a net thermal Hall conductance between the top and the bottom edge [1]}. There is also a net magnon current transport from the hotter to the colder part of the sample [2]}. Within our parameter setting, each sub panel of Fig. REF b represents the \(\kappa _{xy}\)  current corresponding to the dashed energy levels of \(0.637J_1\) , \(1.5J_1\) , and \(2.36J_1\)  in the gap, respectively. As the figure shows, there is a net non-zero current for the left and the right sub panels, but there is no net edge state current in the second gap.
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